Algorithms for Non-Negative Matrix Factorization on Noisy Data With Negative Values
This addresses a specific issue in analyzing noisy astronomical data where negative values arise from noise, offering an incremental improvement for domain applications.
The paper tackles the problem of applying non-negative matrix factorization to noisy data with negative values, which prior methods handled inconsistently, by introducing two algorithms that correctly recover non-negative signals without clipping or offset, as demonstrated numerically.
Non-negative matrix factorization (NMF) is a dimensionality reduction technique that has shown promise for analyzing noisy data, especially astronomical data. For these datasets, the observed data may contain negative values due to noise even when the true underlying physical signal is strictly positive. Prior NMF work has not treated negative data in a statistically consistent manner, which becomes problematic for low signal-to-noise data with many negative values. In this paper we present two algorithms, Shift-NMF and Nearly-NMF, that can handle both the noisiness of the input data and also any introduced negativity. Both of these algorithms use the negative data space without clipping, and correctly recover non-negative signals without any introduced positive offset that occurs when clipping negative data. We demonstrate this numerically on both simple and more realistic examples, and prove that both algorithms have monotonically decreasing update rules.